A344462 Number of pairs (d1,d2) of divisors of n such that d1 < d2 and d2^2 + d1*(n - d2)^2 = n^2.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 7, 1, 3, 3, 4, 1, 7, 1, 5, 3, 3, 1, 9, 2, 3, 3, 5, 1, 9, 1, 5, 3, 3, 3, 10, 1, 3, 3, 7, 1, 9, 1, 5, 5, 3, 1, 11, 2, 5, 3, 5, 1, 9, 3, 7, 3, 3, 1, 13, 1, 3, 5, 6, 3, 9, 1, 5, 3, 7, 1, 13, 1, 3, 5, 5, 3, 9, 1, 9, 4, 3, 1, 13, 3, 3, 3, 7, 1, 13

Offset

1,4

Comments

If p is prime, then a(p) = 1. The only pair of divisors (d1,d2) of p such that d1 < d2 is (1,p), and since p^2 + 1*(p - p)^2 = p^2, (1,p) is the only solution.

Links

Table of n, a(n) for n=1..90.

Formula

a(n) = Sum_{d1|n, d2|n, d1<d2} [d2^2 + d1*(n - d2)^2 = n^2], where [ ] is the Iverson bracket.

Example

a(12) = 7; The 7 pairs are (1,12), (2,4), (2,12), (3,6), (3,12), (4,12), (6,12).

Mathematica

Table[Sum[Sum[KroneckerDelta[k^2 + i*(n - k)^2, n^2] (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]

Crossrefs

Sequence in context: A378181 A353338 A319149 * A321887 A046051 A025812

Adjacent sequences: A344459 A344460 A344461 * A344463 A344464 A344465

Keyword

nonn

Author

Wesley Ivan Hurt, May 19 2021

Status

approved